J. W. Oihhs — Equilibrium of Heterogeneous Substances. 461 



sufficiently small the figure ah e d may be regarded as rectilinear, and 

 that its angles are entirely determined by its tensions. Hence the 

 ratios of u^, Wb, Vc, v-^, for sufficiently small values of ^Jd, are deter- 

 mined by the tensions alone, and for convenience in calculating these 

 ratios, we may suppose jo^j Pb-> Pc to be equal, which will make the 

 figure abed absolutely rectilinear, and make 2^d equal to the other 

 pressures, since it is supposed that this quantity has the value neces- 

 sary for equilibrium. We may obtain a simple expression for the 

 ratios of u^, ^b, ^c, ^d in terms of the tensions in the following 

 manner. We shall write [D B C], [D C A], etc., to denote the areas 

 of triangles having sides equal to the tensions of the surfaces between 

 the masses specified. 



%\ : Vb — triangle b dc: triangle a dc 



: : be sin bed ; ac sin acd 



: : sin bac sin bed : sin abc sin aed 



: : sin y6f:i sin Safd : sin yda sin 6(ia 



: : sin ySft dp : sin ySa 6a 



: : triangle y S f:l: triangle yda 



: : [D B C] : [D C A]. 

 Hence, 



Vj,: v^: Vc-.Vu: :[T>B C] :[D C A]: [I) AB]:[AB C], (634) 

 where 



iV[(C''AB+0'Bc4-^CA)(0-AB+0'BC— 0-ca)(0'bC+0'ca— O-ab) (O'ca+O'aB- O'bc)] 



may be written for [A B C], and analogous expressions for the other 

 symbols, the sign \/ denoting the positive root of the necessarily posi- 

 tive expression which follows. This proportion will hold true in any 

 case of equilibrium, when the tensions satisfy the condition mentioned 

 and Vd is sufficiently small. Now if Pf,=.2)B=-Pc, Pd will have the 

 same value, and w^e shall have by (627) Wy =. 0, and by (633) Ws = 0. 

 But when Vp is very small, the value of W^ is entirely determined by 

 the tensions and Vq. Therefore, whenever the tensions satisfy the 

 condition supposed, and v^, is very small (whether pj,, jOg, j^c are 

 equal or unequal,) 



0= Ws= Wv=pjiVu —PaVa—PbVb-PcVc, (635) 



which with (634) gives 



_ [D B C] p, + [D C A]^B + [D A B]p, 

 2Jn- -[DBC] + [DCA] + [DAB] ' ^ ' 



Since this is the only value oipx, for which equilibrium is possible when 



